| L(s) = 1 | − 2.66·2-s − 1.61·3-s + 5.08·4-s + 2.63·5-s + 4.30·6-s + 0.680·7-s − 8.21·8-s − 0.385·9-s − 7.01·10-s − 2.00·11-s − 8.22·12-s + 4.17·13-s − 1.81·14-s − 4.26·15-s + 11.6·16-s + 0.911·17-s + 1.02·18-s + 5.60·19-s + 13.4·20-s − 1.10·21-s + 5.33·22-s + 4.77·23-s + 13.2·24-s + 1.94·25-s − 11.1·26-s + 5.47·27-s + 3.46·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.933·3-s + 2.54·4-s + 1.17·5-s + 1.75·6-s + 0.257·7-s − 2.90·8-s − 0.128·9-s − 2.21·10-s − 0.604·11-s − 2.37·12-s + 1.15·13-s − 0.484·14-s − 1.10·15-s + 2.92·16-s + 0.221·17-s + 0.241·18-s + 1.28·19-s + 2.99·20-s − 0.240·21-s + 1.13·22-s + 0.994·23-s + 2.71·24-s + 0.389·25-s − 2.17·26-s + 1.05·27-s + 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8233245405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8233245405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 - 0.680T + 7T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 - 0.911T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 0.634T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 8.42T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.423941837668152746411514196478, −7.61189808465700617151255009584, −6.86071803416645524283109851584, −6.19685225130739720609378613198, −5.66309295374661994922551125629, −5.01703101480657056617428352030, −3.23974597096210907114743966683, −2.45500085177614910912906175949, −1.38622712619335533260574633923, −0.76994640986402345057986422562,
0.76994640986402345057986422562, 1.38622712619335533260574633923, 2.45500085177614910912906175949, 3.23974597096210907114743966683, 5.01703101480657056617428352030, 5.66309295374661994922551125629, 6.19685225130739720609378613198, 6.86071803416645524283109851584, 7.61189808465700617151255009584, 8.423941837668152746411514196478
